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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 139650.be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139650.be1 | 139650hv1 | \([1, 1, 0, -13686950, -26651083500]\) | \(-2569823930905/1292464512\) | \(-142612982363618550000000\) | \([]\) | \(15240960\) | \(3.1478\) | \(\Gamma_0(N)\)-optimal |
139650.be2 | 139650hv2 | \([1, 1, 0, 107863675, 334232722125]\) | \(1257792236741495/1165133611008\) | \(-128563049565528883200000000\) | \([]\) | \(45722880\) | \(3.6971\) |
Rank
sage: E.rank()
The elliptic curves in class 139650.be have rank \(1\).
Complex multiplication
The elliptic curves in class 139650.be do not have complex multiplication.Modular form 139650.2.a.be
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.